Step 1 — Recall the diagonal formula for a parallelogram.

If $\vec{a}$ and $\vec{b}$ are the two adjacent sides, then the diagonals are represented by

$$\vec{d_1} = \vec{a} + \vec{b} \qquad \text{and} \qquad \vec{d_2} = \vec{a} - \vec{b}$$

Step 2 — Find the first diagonal.

$$\vec{d_1} = (3\hat{i} - 2\hat{j} + 2\hat{k}) + (\hat{i} + 0\hat{j} + 2\hat{k})$$

$$\vec{d_1} = 4\hat{i} - 2\hat{j} + 4\hat{k}$$

Step 3 — Find the length of the first diagonal.

$$|\vec{d_1}| = \sqrt{4^2 + (-2)^2 + 4^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6$$

Step 4 — Find the second diagonal.

$$\vec{d_2} = (3\hat{i} - 2\hat{j} + 2\hat{k}) - (\hat{i} + 0\hat{j} + 2\hat{k})$$

$$\vec{d_2} = 2\hat{i} - 2\hat{j} + 0\hat{k}$$

Step 5 — Find the length of the second diagonal.

$$|\vec{d_2}| = \sqrt{2^2 + (-2)^2 + 0^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$$

Final answer:

$$\boxed{\vec{d_1} = 4\hat{i} - 2\hat{j} + 4\hat{k},\ |\vec{d_1}| = 6 \ ;\quad \vec{d_2} = 2\hat{i} - 2\hat{j},\ |\vec{d_2}| = 2\sqrt{2}}$$